Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

P. Z. Mkrtychyan

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 178-190

Résumé

In a bounded domain of the n-dimensional $$ \sum_{i=1}^n\frac\partial{\partial x_i}(a^{l_i}(u)|u_{x_i}|^{m_i-2}u_{x_i})=f(x), $$ where $x=(x_1,\dots,x_n)$, $l_i\ge0$, $m_i>1$, the function $f$ is summable with some power, the nonnegative continuous function $a(u)$ vanishes at a finite number of points and satisfies $\varliminf_{|u|\to\infty}a(u)>0$. One proves the existence of bounded generalized solutions with a finite integral $$ \int_\Omega\sum_{i=1}^na^{l_i}(u)|u_{x_i}|^{m_i}\,dx $$ of the Dirichlet problem with zero boundary conditions.